Derivation of Feynman–Kac and Bloch–Torrey Equations in a Trapping Medium

Abstract

We derive rigorously the fractional counterpart of the Feynman–Kac equation for a transport problem with trapping events characterized by fat-tailed time distributions. Our starting point is a random walk model with an inherent time subordination process due to the difference between clock times and operational times. We pass to the hydrodynamic limit using weak convergence arguments. Due to the lack of regularity of physical data, we use the framework of measure theory. We finally derive a Bloch–Torrey type equation for nuclear magnetic resonance data in this subdiffusive context.

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Appendix

Sequential Compactness of \((\tau b_{m}^{\tau })_{\tau >0}\)

We prove here that the definition of the renewal measures \(b_{m}^{\tau }\) let us ensure that the sequence \((\tau b_{m}^{\tau })_{\tau >0}\) contains some narrowly convergent subsequence. We have the following result.

Lemma 2

Proof

Due to Eq. 10, \((\tau b_{m}^{\tau })\) is uniformly bounded in \(\mathcal {M}_{b}^{+}(E)\). We thus assert that \((\tau b_{m}^{\tau })\subset \mathcal {M}_{b}^{+}(E)\) is vaguely relatively compact (i.e. for the topology defined by continuous test functions with compact support in E). Space E being a Polish space, E is strongly Prokhorov (in the sense of Fremlin 2003): any vaguely compact subset of signed tight measures on E is uniformly tight, and thus narrowly relatively compact in \(\mathcal {M}_{b}^{+}(E)\). Hence it remains to check that \(\tau b_{m}^{\tau }\) is a tight measure: for any 𝜖 > 0, there exists some compact subset K𝜖 ⊂ E such that

$$ b_{m}^{\tau} (E \setminus K_{\epsilon}) \le \epsilon. $$

(49)

We bear in mind that the number of renewals during \([0,\bar {t}_{m}]\) is at most \(N + 1=\lfloor \frac {\bar {t}_{m}}{\tau } \rfloor + 1\). Result (49) is ensured by choosing

Indeed, the control by τ/2 in Eq. 51 ensures that there is at most one renewal in \([\bar {t}_{m}-\eta _{2}(\epsilon ),\bar {t}_{m}]\). Moreover, provided \(\tau <\bar {t}_{m}\), the rank of this renewal is between 1 and N, and it occurs with a probability of at most \({\Sigma }_{n = 1}^{N}\Vert \psi _{\tau ,\gamma }\Vert _{\infty }\eta _{2}(\epsilon )\), with ψτ,γ(t) = τ− 1/γψγ(τ− 1/γt). We thus impose the control by 𝜖/2 in Eq. 51. We finally set

An Auxiliary Convergence Result

The previous subsection gives a compactness result for the narrow topology. A critical difference between such a weak convergence result, \(\mu ^{\tau } \rightharpoonup \mu \), and a strong convergence result in \(\mathcal {M}^{+}_{b}(E)\), is that we cannot ensure the convergence of μτ(A) to μ(A) for any subset A ⊂ E. Yet we have the following result.

Once again, we conclude that the limit of the latter quantity is zero by using the uniform continuity of g (for the limit of the first term in the right hand side), and by using estimate \(\tau b_{m}^{\tau }(\mathbb {R}\times J^{\tau } \times \mathbb {R})\le \tau \) (for the limit of the second term in the right hand side). We have proved that \(E_{+,i}^{\tau } - E_{-,i}^{\tau } \ {\overset {\mathcal {D}}{\rightharpoonup }}\ 0\), i = 1, 2.